On the comoving distance as arc-length in four dimensions

Abstract

The inner product provides a conceptually and algorithmically simple method for cal- culating the comoving distance between two cosmological objects given their redshifts, right ascension and declination, and arbitrary constant curvature. The key to this is that just as a distance between two points ‘on’ the surface of the ordinary 2-sphere S2 is simply an arc-length (angle multiplied by radius) in ordinary Euclidean 3-space E3, the distance between two points ‘on’ a 3-sphere S3 (a 3-hyperboloid H3) is simply an ‘arc-length’ in Euclidean 4-space E4 (Minkowski 4-spaceM4), i.e. an ‘hyper-angle’ multiplied by the curvature radius of the 3-sphere (3-hyperboloid).